Monoidal t-norm logic

In fuzzy logic, rather than regarding statements as being either true or false, we associate each statement with a numerical confidence in that statement. By convention the confidences range over the unit interval [0,1], where the maximal confidence 1 corresponds to the classical concept of true and the minimal confidence 0 corresponds to the classical concept of false. T-norms are binary functions on the real unit interval [0, 1] that in fuzzy logic are often used to represent a conjunction connective; if a,b \in [0,1] are the confidences we ascribe to the statements A and B respectively, then one uses a t-norm * to calculate the confidence a*b ascribed to the compound statement ‘A and B’. A t-norm * has to satisfy the properties of :commutativity a*b = b*a , :associativity (a*b)*c = a*(b*c) , :monotonicity — if a \leqslant b and c \leqslant d then a*c \leqslant b*d , :and having 1 as identity element 1*a = a . Notably absent from this list is the property of idempotence a*a = a ; the closest one gets is that a*a \leqslant 1*a = a . It may seem strange to be less confident in ‘A and A’ than in just A, but we generally do want to allow for letting the confidence a*b in a combined ‘A and B’ be less than both the confidence a in A and the confidence b in B, and then the ordering a*b by monotonicity requires a*a \leqslant a*b . Another way of putting it is that the t-norm can only take into account the confidences as numbers, not the reasons that may be behind ascribing those confidences; thus it cannot treat ‘A and A’ differently from ‘A and B, where we are equally confident in both’. Because the symbol \wedge via its use in lattice theory is very closely associated with the idempotence property, it can be useful to switch to a different symbol for conjunction that is not necessarily idempotent. In the fuzzy logic tradition one sometimes uses \& for this "strong" conjunction, but this article follows the substructural logic tradition of using \otimes for the strong conjunction; thus a*b is the confidence we ascribe to the statement A \otimes B (still read ‘A and B’, perhaps with ‘strong’ or ‘multiplicative’ as qualification of the ‘and’). Having formalised conjunction \otimes, one wishes to continue with the other connectives. One approach to doing that is to introduce negation as an order-reversing map [0,1] \longrightarrow [0,1], then defining remaining connectives using De Morgan's laws, material implication, and the like. A problem with doing so is that the resulting logics may have undesirable properties: they may be too close to classical logic, or if not conversely not support expected inference rules. An alternative that makes the consequences of different choices more predictable is to instead continue with implication \to as the second connective: this is overall the most common connective in axiomatisations of logic, and it has closer ties to the deductive aspects of logic than most other connectives. A confidence counterpart \Rightarrow of the traditional implication connective may in fact be defined directly as the residuum of the t-norm. The logical link between conjunction and implication is provided by something as fundamental as the inference rule modus ponens A, A \to B \vdash B: from A and A \to B follows B. In the fuzzy logic case that is more rigorously written as A \otimes (A \to B) \vdash B, because this makes explicit that our confidence for the premise(s) here is that in A \otimes (A \to B), not those in A and A \to B separately. So if a and b are our confidences in A and B respectively, then a \Rightarrow b is the sought confidence in A \to B, and a * (a \Rightarrow b) is the combined confidence in A \otimes (A \to B). We require that : a * (a \mathbin{\Rightarrow} b) \leqslant b since our confidence b for B should not be less than our confidence a * (a \Rightarrow b) in the statement A \otimes (A \to B) from which B logically follows. This bounds the sought confidence a \Rightarrow b, and one approach for turning \Rightarrow into a binary operation like * would be to make it as large as possible while respecting this bound: : a \mathbin{\Rightarrow} b \equiv \sup \left\{ x \in [0,1] \;\big|\; a*x \leqslant b \right\} . Taking x=0 gives a*x = a*0 \leqslant 1*0 = 0 \leqslant b , so the supremum is always of a nonempty bounded set and thus well-defined. For a general t-norm there remains the possibility that f_a(x) = a*x has a jump discontinuity at x = a \mathbin{\Rightarrow} b , in which case a * (a \mathbin{\Rightarrow} b) could come out strictly larger than b even though a \mathbin{\Rightarrow} b is defined as the least upper bound of xs satisfying a*x \leqslant b ; to prevent that and have the construction work as expected, we require that the t-norm * is left-continuous. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. More algebraically, we say that an operation \Rightarrow is a residuum of a t-norm * if for all a, b, and c it satisfies : a*b\le c if and only if a\le (b \mathbin{\Rightarrow} c). This equivalence of numerical comparisons mirrors the equivalence of entailments : A \otimes B \vdash C if and only if A \vdash B \to C that exists because any proof of C from the premise A \otimes B can be converted into a proof of B \to C from the premise A by doing an extra implication introduction step, and conversely any proof of B \to C from the premise A can be converted into a proof of C from the premise A \otimes B by doing an extra implication elimination step. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation \neg x=(x\mathbin{\Rightarrow} 0). In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section Standard semantics below) determine the truth values of complex propositional formulae in [0, 1]. Formulae that always evaluate to 1 are then called tautologies with respect to the given left-continuous t-norm *, or *\mbox{-}tautologies. The set of all *\mbox{-}tautologies is called the logic of the t-norm *, since these formulae represent the laws of fuzzy logic (determined by the t-norm) that hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to all left-continuous t-norms: they represent general laws of propositional fuzzy logic that are independent of the choice of a particular left-continuous t-norm. These formulae form the logic MTL, which can thus be characterized as the logic of left-continuous t-norms. == Syntax ==